Proposition 2.1  ([1]}) Assuming that the initial data \(u(0,x) = u_0(x)\in H^2(\Omega )\bigcap H_0^1(\Omega )\) .
Then, for \(\alpha \in (0,1)\)  and \(T>0\) , there exists a unique solution to the system (REF ) with homogeneous Dirichlet boundary condition satisfying
\( & u\in H^\alpha (0,T;L^2(\Omega ))\bigcap L^2(0,T;H^2(\Omega )\cap H_0^1(\Omega )),~u\in L^\infty ((0,T)\times \Omega ),\nonumber \\&u\in C^\alpha ([0,T]; L^2(\Omega ))\bigcap C([0,T];H^2(\Omega )\cap H_0^1(\Omega )),~ \partial _t^\alpha u\in C([0,T];L^2(\Omega )),\nonumber \\& \partial _t u\in C([0,T]; L^2(\Omega )),~\Vert \partial _tu(t)\Vert \le Ct^{\alpha -1} \text{ for } t\in (0,T].\nonumber \) 
